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Denote by $\mathcal P(S)$ the semigroup obtained by endowing the non-empty subsets of a "ground semigroup" $S$ (written multiplicatively) with the operation of setwise multiplication induced by $S$: $$ (X, Y) \mapsto XY := \{xy \colon x \in X, \, y \in Y\}. $$ In the literature, $\mathcal P(S)$ is called the power semigroup (or global) of $S$. Two semigroups are then called globally isomorphic if their power semigroups are isomorphic.

It is easy to show that two groups $G$ and $G'$ are globally isomorphic iff they are semigroup-isomorphic (and hence group-isomorphic). In fact, something stronger is true: Every semigroup isomorphism from $\mathcal P(G)$ to $\mathcal P(G')$ restricts (with a little abuse of language) to a group isomorphism from $G$ to $G'$. The basic point is simply that

  1. the group of units of the power semigroup of a monoid $M$ is isomorphic to the group of units of $M$, and
  2. every semigroup isomorphism from a monoid to another is, a fortiori, a monoid isomorphism (that is, maps the identity to the identity).

In the introduction of [Gould, Iskra, and Tsinakis, Globally determined classes of semigroups, Semigroup Forum 29 (1984) 365-374], the authors write, "In 1967-68 Tamura and Shafer noted that the class of all groups [16] is globally determined" (emphasis mine). Here, [16] is [Tamura and Shafer, Power semigroups, Math. Japon. 12 (1967), 25-32]. I've tried to get my hands on the latter paper, but to no avail. So, I don't know exactly what is inside it. But I seem to gather from McAlister's zbMATH review (here) that Tamura and Shafer only deal with the case of finite groups. This leads me to the following:

Question(s). What do Tamura and Shafer really prove in their paper? Is their proof critically depending on the finiteness of the groups involved? Does their proof also cover the case where at least one of $H$ and $K$ is a group?

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    $\begingroup$ It appears that Shafer published a 1 page note in the same issue (on page 32) handling general groups. Shafer, J. Note on power semigroups. Math. Japon. 12 (1967), 32. My guess is they somehow missed something obvious when first writing the paper and then figured out the correct result before it appeared. But I also cannot access the paper. $\endgroup$
    – Benjamin Steinberg
    Commented Oct 15, 2023 at 20:11
  • $\begingroup$ Thank you, Ben, I was not aware of Shafer's note. As for the last question in the OP, there is a comment from Oct 2012 that sounds relevant: mathoverflow.net/questions/110412/#comment284325_110421 (the comment is essentially asking if a group can only be globally isomorphic to another group, and in a subsequent comment it is explained how the answer is yes if the group is finite). $\endgroup$
    – Salvo Tringali
    Commented Oct 16, 2023 at 6:47

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Thanks to Valentin Havlovec (TU Graz, Austria), I've finally got a copy of (i) Tamura and Shafer's paper [Power Semigroups, Math. Japon. 12 (1967), 25-32] and (ii) the note by Shafer mentioned by Benjamin Steinberg in a comment to the OP.

As guessed by Benjamin in his comment, Tamura and Shafer must have somehow overlooked something obvious when writing the paper. In fact, Shafer's (less-than-one-page) note appears immediately after the joint paper with Tamura in the same issue of Math. Japon., and Shafer writes, "By using a method suggested to the author by T.J. Head, it is possible to prove Theorem 2 of Power Semigroups [2] without the restriction to finite groups".

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